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Let $a$ and $b$ be two integers. The relation $\alpha\subseteq \mathbb{Z}\times\mathbb{Z}$ is defined by $x \alpha y$ if and only if $аx+by=1$.

Find all possible $a$ and $b$ for which $α$ is nonempty.

For all $а$ and $b$ proof existence or absence of each of following properties: irreflexive, symmetric, antysymmetric and transitive.

Here is my attempt [link]

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    Do you know Bezouts identity? It shows directly that $(a,b)\in\alpha$ if and only if $\operatorname{gcd}(a,b)=1$. – 2017-01-04
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    Sorry, I don't know that identity – 2017-01-04
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    @FateMetric: [Bezout’s identity](https://en.wikipedia.org/wiki/B%C3%A9zout's_identity). – 2017-01-04

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